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Properties

Label 2-637-91.45-c1-0-29
Degree $2$
Conductor $637$
Sign $0.198 + 0.980i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.56 + 1.56i)2-s + (−0.959 + 0.554i)3-s − 2.91i·4-s + (0.784 − 2.92i)5-s + (0.635 − 2.37i)6-s + (1.43 + 1.43i)8-s + (−0.885 + 1.53i)9-s + (3.36 + 5.82i)10-s + (−0.188 + 0.705i)11-s + (1.61 + 2.79i)12-s + (2.65 + 2.44i)13-s + (0.869 + 3.24i)15-s + 1.33·16-s − 3.24·17-s + (−1.01 − 3.79i)18-s + (−3.85 + 1.03i)19-s + ⋯
L(s)  = 1  + (−1.10 + 1.10i)2-s + (−0.554 + 0.319i)3-s − 1.45i·4-s + (0.351 − 1.31i)5-s + (0.259 − 0.968i)6-s + (0.506 + 0.506i)8-s + (−0.295 + 0.511i)9-s + (1.06 + 1.84i)10-s + (−0.0569 + 0.212i)11-s + (0.466 + 0.807i)12-s + (0.735 + 0.678i)13-s + (0.224 + 0.838i)15-s + 0.334·16-s − 0.787·17-s + (−0.239 − 0.894i)18-s + (−0.885 + 0.237i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.198 + 0.980i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174254 - 0.142461i\)
\(L(\frac12)\) \(\approx\) \(0.174254 - 0.142461i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-2.65 - 2.44i)T \)
good2 \( 1 + (1.56 - 1.56i)T - 2iT^{2} \)
3 \( 1 + (0.959 - 0.554i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.784 + 2.92i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.188 - 0.705i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 3.24T + 17T^{2} \)
19 \( 1 + (3.85 - 1.03i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 5.96iT - 23T^{2} \)
29 \( 1 + (-2.78 + 4.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.99 - 1.07i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.97 + 6.97i)T + 37iT^{2} \)
41 \( 1 + (2.46 - 0.660i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.73 - 3.30i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.69 - 0.454i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.37 + 5.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.33 + 2.33i)T - 59iT^{2} \)
61 \( 1 + (6.30 + 3.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.27 + 1.68i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.06 + 2.16i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.62 + 6.08i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.87 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.4 + 10.4i)T + 83iT^{2} \)
89 \( 1 + (-6.62 + 6.62i)T - 89iT^{2} \)
97 \( 1 + (2.37 - 8.86i)T + (-84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.21150844418123137312478061076, −9.152790984960248244431182353960, −8.670214714813207277308858132685, −8.057384570310287649242074121636, −6.75161735112123816137171959553, −6.04752192214796445890189558057, −5.13577744068767143239711267621, −4.28206761725442836215790025038, −1.84321692746835900234507783640, −0.19079501419507387860050741829, 1.50407852011999449181907007396, 2.81987281090599886169829595762, 3.56121497369915332438547594165, 5.56386873821108457149793300821, 6.46509457590775793944856961825, 7.25953118701441801787105538463, 8.510370589453663048280071818020, 9.115152080544557678070682338950, 10.33167549649323890752518954818, 10.62503904213656545132221526808

Graph of the $Z$-function along the critical line